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WHAT KATZ ACTUALLY SAID “we assume that each link independently has the same probability of being effective” … “we conceive a constant , depending on the group and the context of the particular investigation, which has the force of a probability of effectiveness of a single link. A k-step chain then, has probability of being effective.” “We wish to find the column sums of the matrix” Leo Katz 1953, A New Status Index Derived from Sociometric Analysis, Psychometria 18(1):39-43David F. Gleich (Sandia) Berkeley SCMC Seminar 8 of 43

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A MODERN TAKEThe Katz score (node-based) is The Katz score (edge-based) is David F. Gleich (Sandia) Berkeley SCMC Seminar 9 of 43

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Carl Neumann I’ve heard the Neumann series called the “von Neumann” series more than I’d like! In fact, the von Neumann kernel of a graph should be named the “Neumann” kernel! Wikipedia pageDavid F. Gleich (Sandia) Berkeley SCMC Seminar 11 / 43

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WHAT DO OTHER PEOPLE DO?1) Just work with the linear algebra formulations2) For Katz, Truncate the Neumann series as a few (3-5) terms3) Use low-rank approximations from EVD(A) or EVD(L)4) For commute, use Johnson-Lindenstrauss inspired random sampling5) Approximately decompose into smaller problems Liben-Nowell and Kleinberg CIKM2003, Acar et al. ICDM2009, Spielman and Srivastava STOC2008, Sarkar and Moore UAI2007,Wang et al. ICDM2007David F. Gleich (Sandia) Berkeley SCMC Seminar 16 of 43

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THE PROBLEM All of these techniques are preprocessing based because most people’s goal is to compute all the scores. We want to avoid preprocessing the graph. There are a few caveats here! i.e. one could solve the system instead of looking for the matrix inverseDavid F. Gleich (Sandia) Berkeley SCMC Seminar 17 of 43

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WHY NO PREPROCESSING? The graph is constantly changing as I rate new movies.David F. Gleich (Sandia) Berkeley SCMC Seminar 18 of 43

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MMQ PROCEDUREGoal Given 1. Run k-steps of Lanczos on starting with 2. Compute , with an additional eigenvalue at , set Correspond to a Gauss-Radau rule, with u as a prescribed node3. Compute , with an additional eigenvalue at , set Correspond to a Gauss-Radau rule, with l as a prescribed node4. Output as lower and upper bounds on David F. Gleich (Sandia) Berkeley SCMC Seminar 24 of 43

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THE ALGORITHM FOR KATZFor Init: Pick as max Storing the non-zeros of the residual in a heap makes picking the max log(n) time. See Anderson et al. FOCS2008 for moreDavid F. Gleich (Sandia) Berkeley SCMC Seminar 34 of 43

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CONVERGENCE?The “standard” proof works for 1/max-degree.For , then for symmetric ,this algorithm is the Gauss-Southwellprocedure and it still converges.David F. Gleich (Sandia) Berkeley SCMC Seminar 35 of 43

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WARTS AND TODOSStopping criteria on our top-k algorithmcan be a bit hairy… we should refine it.The top-k approach doesn’t work right forcommute time… we have an alternative. requires the entire diagonal of Take advantage of new research to “seed”commute time better! von Luxburg et al. NIPS2010David F. Gleich (Sandia) Berkeley SCMC Seminar 42 of 43